scholarly journals The method of upper, lower solutions and monotone iterative scheme for higher order hyperbolic partial differential equations

1989 ◽  
Vol 47 (1) ◽  
pp. 153-170 ◽  
Author(s):  
Ravi P. Agarwal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Iterative Scheme ◽  
Initial Conditions ◽  
Upper And Lower Solutions ◽  
Hyperbolic Partial Differential Equations ◽  
Partial Differential ◽  
Monotone Iterative ◽  
Variation Of Parameters ◽  
Variation Of Parameters Formula

AbstractUniformly monotone convergent iterative methods for obtaining multiple solutions of (n + m)th order hyperbolic partial differential equations together with initial conditions are discussed. Appropriate partial differential inequalities which connect upper and lower solutions, and variation of parameters formula is developed.

scholarly journals Exponential Jacobi spectral method for hyperbolic partial differential equations

2019 ◽  
Vol 13 (4) ◽  
pp. 347-354 ◽  
Author(s):  
Y. H. Youssri ◽  
R. M. Hafez
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Collocation Method ◽  
Initial Conditions ◽  
Operational Matrix ◽  
Hyperbolic Partial Differential Equations ◽  
Partial Differential ◽  
Linear System Of Equations ◽  
Operational Matrix Of Differentiation ◽  
Jacobi Functions

Abstract Herein, we have proposed a scheme for numerically solving hyperbolic partial differential equations (HPDEs) with given initial conditions. The operational matrix of differentiation for exponential Jacobi functions was derived, and then a collocation method was used to transform the given HPDE into a linear system of equations. The preferences of using the exponential Jacobi spectral collocation method over other techniques were discussed. The convergence and error analyses were discussed in detail. The validity and accuracy of the proposed method are investigated and checked through numerical experiments.

Key words: Nonlinear Differential-Difference Equations; Exp-Function Method; N-Soliton Solutions

2010 ◽  
Vol 65 (11) ◽  
pp. 935-949 ◽  
Author(s):  
Mehdi Dehghan ◽  
Jalil Manafian ◽  
Abbas Saadatmandi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Analysis Method ◽  
Partial Differential ◽  
Integer Order ◽  
Homotopy Analysis

In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the homotopy analysis method for partial differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

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